Yo’av Rieck and Yasushi Yamashita

Abstract

We view closed orientable 3–manifolds
as covers of S3 branched over
hyperbolic links. To a cover M→pS3,
of degree p and branched over
a hyperbolic link L⊂S3, we assign
the complexity pVol(S3∖L). We define an
invariant of 3–manifolds, called
the link volume and denoted by LinkVol(M),
that assigns to a 3-manifold M
the infimum of the complexities of all possible covers
M→S3, where the only
constraint is that the branch set is a hyperbolic link. Thus the link volume measures how efficiently
M can be represented
as a cover of S3.

We study the basic properties of the link volume and related
invariants, in particular observing that for any hyperbolic manifold
M,
Vol(M) is less
than LinkVol(M). We
prove a structure theorem that is similar to (and uses) the celebrated theorem of Jørgensen
and Thurston. This leads us to conjecture that, generically, the link volume of a hyperbolic
3–manifold
is much bigger than its volume.

Finally we prove that the link volumes of the manifolds obtained by Dehn filling a
manifold with boundary tori are linearly bounded above in terms of the length of the
continued fraction expansion of the filling curves.